Improved lower and upper bounds for LCD codes?
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Linear complementary dual (LCD) codes are linear codes which intersect their dual codes trivially, which have been of interest and extensively studied due to its wide applications. In this paper, we give some methods for constructing LCD codes over small fields by modifying some known methods. We show that all odd-like binary LCD codes, ternary LCD codes and quaternary Hermitian LCD codes can be constructed by the modified methods. Using these methods, we construct a lot of optimal binary LCD codes, ternary LCD codes and quaternary Hermitian LCD codes, which improve the known lower bounds on the largest minimum weights. Furthermore, we give two counterexamples to show that the conjecture proposed by Bouyuklieva (Des. Codes Cryptogr. 89(11): 2445-2461, 2021) is invalid.
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